Intution behind the gradient giving the steepest ascent in 2D

[Mohankpvk]

Can this(image) be used as a proof that the direction of gradient gives the direction of steepest ascent(in 2D).Am I understanding it right ?.The function 'f' in the image is a scalar valued function.Please explain.

 

[jedishrfu]

This article from Better Explained may help you understand it better:

https://betterexplained.com/articles/vector-calculus-understanding-the-gradient/

Khan Academy has some videos too:

https://www.khanacademy.org/math/mu...-gradient-is-the-direction-of-steepest-ascent

 

[mathwonk]

the gradient is a vector whose dot product with a given direction vector gives the rate of change of the function in that direction. hence it dots to zero along a direction where the function is constant, and hence the gradient is perpendicular to the level curves (for a function defined on the plane) of the function. It seems obvious that the direction in which the function increases fastest is perpendicular to the level curve through a given point. just imagine you are walking along a level path cut into the side of a hill, wouldn't the slope of the hill be greatest in a direction perpendicular to the path?

 

[bigfooted]

Here's a nice (the best actually) introduction to steepest descent and conjugate gradient methods:

https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf